Theorem filconn 23799

Description: A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
filconn	⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)

Proof of Theorem filconn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2		filunibas 23797	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
3	2	fveq2d 6832	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (Fil‘∪ 𝐹) = (Fil‘𝑋))
4	1, 3	eleqtrrd 2836	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (Fil‘∪ 𝐹))
5		nss 3995	. . . . . . . 8 ⊢ (¬ 𝑥 ⊆ {∅} ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅}))
6		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝐹 ∈ (Fil‘∪ 𝐹))
7		ssel2 3925	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ⊆ (𝐹 ∪ {∅}) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝐹 ∪ {∅}))
8	7	adantll 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝐹 ∪ {∅}))
9		elun 4102	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (𝐹 ∪ {∅}) ↔ (𝑦 ∈ 𝐹 ∨ 𝑦 ∈ {∅}))
10	8, 9	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∈ 𝐹 ∨ 𝑦 ∈ {∅}))
11	10	orcomd 871	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∈ {∅} ∨ 𝑦 ∈ 𝐹))
12	11	ord 864	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦 ∈ 𝑥) → (¬ 𝑦 ∈ {∅} → 𝑦 ∈ 𝐹))
13	12	impr 454	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑦 ∈ 𝐹)
14		uniss 4866	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ⊆ ∪ (𝐹 ∪ {∅}))
15	14	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → ∪ 𝑥 ⊆ ∪ (𝐹 ∪ {∅}))
16		uniun 4881	. . . . . . . . . . . . . 14 ⊢ ∪ (𝐹 ∪ {∅}) = (∪ 𝐹 ∪ ∪ {∅})
17		0ex 5247	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ V
18	17	unisn 4877	. . . . . . . . . . . . . . 15 ⊢ ∪ {∅} = ∅
19	18	uneq2i 4114	. . . . . . . . . . . . . 14 ⊢ (∪ 𝐹 ∪ ∪ {∅}) = (∪ 𝐹 ∪ ∅)
20		un0 4343	. . . . . . . . . . . . . 14 ⊢ (∪ 𝐹 ∪ ∅) = ∪ 𝐹
21	16, 19, 20	3eqtrri 2761	. . . . . . . . . . . . 13 ⊢ ∪ 𝐹 = ∪ (𝐹 ∪ {∅})
22	15, 21	sseqtrrdi 3972	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → ∪ 𝑥 ⊆ ∪ 𝐹)
23		elssuni 4889	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑥 → 𝑦 ⊆ ∪ 𝑥)
24	23	ad2antrl 728	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑦 ⊆ ∪ 𝑥)
25		filss 23769	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ ∪ 𝑥 ⊆ ∪ 𝐹 ∧ 𝑦 ⊆ ∪ 𝑥)) → ∪ 𝑥 ∈ 𝐹)
26	6, 13, 22, 24, 25	syl13anc 1374	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → ∪ 𝑥 ∈ 𝐹)
27		elun1 4131	. . . . . . . . . . 11 ⊢ (∪ 𝑥 ∈ 𝐹 → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
28	26, 27	syl 17	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
29	28	ex 412	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → ((𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
30	29	exlimdv 1934	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → (∃𝑦(𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
31	5, 30	biimtrid 242	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → (¬ 𝑥 ⊆ {∅} → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
32		uni0b 4884	. . . . . . . 8 ⊢ (∪ 𝑥 = ∅ ↔ 𝑥 ⊆ {∅})
33		ssun2 4128	. . . . . . . . . 10 ⊢ {∅} ⊆ (𝐹 ∪ {∅})
34	17	snid 4614	. . . . . . . . . 10 ⊢ ∅ ∈ {∅}
35	33, 34	sselii 3927	. . . . . . . . 9 ⊢ ∅ ∈ (𝐹 ∪ {∅})
36		eleq1 2821	. . . . . . . . 9 ⊢ (∪ 𝑥 = ∅ → (∪ 𝑥 ∈ (𝐹 ∪ {∅}) ↔ ∅ ∈ (𝐹 ∪ {∅})))
37	35, 36	mpbiri 258	. . . . . . . 8 ⊢ (∪ 𝑥 = ∅ → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
38	32, 37	sylbir 235	. . . . . . 7 ⊢ (𝑥 ⊆ {∅} → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
39	31, 38	pm2.61d2 181	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
40	39	ex 412	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
41	40	alrimiv 1928	. . . 4 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
42		filin 23770	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ 𝐹)
43		elun1 4131	. . . . . . . . . 10 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝐹 → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
44	42, 43	syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
45	44	3expa 1118	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
46	45	ralrimiva 3125	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
47		elsni 4592	. . . . . . . . 9 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
48		ineq2 4163	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑥 ∩ 𝑦) = (𝑥 ∩ ∅))
49		in0 4344	. . . . . . . . . . 11 ⊢ (𝑥 ∩ ∅) = ∅
50	48, 49	eqtrdi 2784	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑥 ∩ 𝑦) = ∅)
51	50, 35	eqeltrdi 2841	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
52	47, 51	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ {∅} → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
53	52	rgen 3050	. . . . . . 7 ⊢ ∀𝑦 ∈ {∅} (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅})
54		ralun 4147	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}) ∧ ∀𝑦 ∈ {∅} (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅})) → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
55	46, 53, 54	sylancl 586	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
56	55	ralrimiva 3125	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
57		elsni 4592	. . . . . . 7 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
58		ineq1 4162	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑥 ∩ 𝑦) = (∅ ∩ 𝑦))
59		0in 4346	. . . . . . . . . 10 ⊢ (∅ ∩ 𝑦) = ∅
60	58, 59	eqtrdi 2784	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ∩ 𝑦) = ∅)
61	60, 35	eqeltrdi 2841	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
62	61	ralrimivw 3129	. . . . . . 7 ⊢ (𝑥 = ∅ → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
63	57, 62	syl 17	. . . . . 6 ⊢ (𝑥 ∈ {∅} → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
64	63	rgen 3050	. . . . 5 ⊢ ∀𝑥 ∈ {∅}∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅})
65		ralun 4147	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}) ∧ ∀𝑥 ∈ {∅}∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅})) → ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
66	56, 64, 65	sylancl 586	. . . 4 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
67		p0ex 5324	. . . . . 6 ⊢ {∅} ∈ V
68		unexg 7682	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ {∅} ∈ V) → (𝐹 ∪ {∅}) ∈ V)
69	67, 68	mpan2 691	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝐹 ∪ {∅}) ∈ V)
70		istopg 22811	. . . . 5 ⊢ ((𝐹 ∪ {∅}) ∈ V → ((𝐹 ∪ {∅}) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})) ∧ ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))))
71	69, 70	syl 17	. . . 4 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ((𝐹 ∪ {∅}) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})) ∧ ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))))
72	41, 66, 71	mpbir2and 713	. . 3 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝐹 ∪ {∅}) ∈ Top)
73	21	cldopn 22947	. . . . . . . 8 ⊢ (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → (∪ 𝐹 ∖ 𝑥) ∈ (𝐹 ∪ {∅}))
74		elun 4102	. . . . . . . 8 ⊢ ((∪ 𝐹 ∖ 𝑥) ∈ (𝐹 ∪ {∅}) ↔ ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 ∨ (∪ 𝐹 ∖ 𝑥) ∈ {∅}))
75	73, 74	sylib 218	. . . . . . 7 ⊢ (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 ∨ (∪ 𝐹 ∖ 𝑥) ∈ {∅}))
76		elun 4102	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹 ∪ {∅}) ↔ (𝑥 ∈ 𝐹 ∨ 𝑥 ∈ {∅}))
77		filfbas 23764	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → 𝐹 ∈ (fBas‘∪ 𝐹))
78		fbncp 23755	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (fBas‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ¬ (∪ 𝐹 ∖ 𝑥) ∈ 𝐹)
79	77, 78	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ¬ (∪ 𝐹 ∖ 𝑥) ∈ 𝐹)
80	79	pm2.21d 121	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅))
81	80	ex 412	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ∈ 𝐹 → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅)))
82	57	a1i13 27	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ∈ {∅} → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅)))
83	81, 82	jaod 859	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ((𝑥 ∈ 𝐹 ∨ 𝑥 ∈ {∅}) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅)))
84	76, 83	biimtrid 242	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ∈ (𝐹 ∪ {∅}) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅)))
85	84	imp 406	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅))
86		elsni 4592	. . . . . . . . 9 ⊢ ((∪ 𝐹 ∖ 𝑥) ∈ {∅} → (∪ 𝐹 ∖ 𝑥) = ∅)
87		elssuni 4889	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐹 ∪ {∅}) → 𝑥 ⊆ ∪ (𝐹 ∪ {∅}))
88	87, 21	sseqtrrdi 3972	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐹 ∪ {∅}) → 𝑥 ⊆ ∪ 𝐹)
89	88	adantl 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → 𝑥 ⊆ ∪ 𝐹)
90		ssdif0 4315	. . . . . . . . . . 11 ⊢ (∪ 𝐹 ⊆ 𝑥 ↔ (∪ 𝐹 ∖ 𝑥) = ∅)
91	90	biimpri 228	. . . . . . . . . 10 ⊢ ((∪ 𝐹 ∖ 𝑥) = ∅ → ∪ 𝐹 ⊆ 𝑥)
92		eqss 3946	. . . . . . . . . . 11 ⊢ (𝑥 = ∪ 𝐹 ↔ (𝑥 ⊆ ∪ 𝐹 ∧ ∪ 𝐹 ⊆ 𝑥))
93	92	simplbi2 500	. . . . . . . . . 10 ⊢ (𝑥 ⊆ ∪ 𝐹 → (∪ 𝐹 ⊆ 𝑥 → 𝑥 = ∪ 𝐹))
94	89, 91, 93	syl2im 40	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → ((∪ 𝐹 ∖ 𝑥) = ∅ → 𝑥 = ∪ 𝐹))
95	86, 94	syl5 34	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → ((∪ 𝐹 ∖ 𝑥) ∈ {∅} → 𝑥 = ∪ 𝐹))
96	85, 95	orim12d 966	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 ∨ (∪ 𝐹 ∖ 𝑥) ∈ {∅}) → (𝑥 = ∅ ∨ 𝑥 = ∪ 𝐹)))
97	75, 96	syl5 34	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → (𝑥 = ∅ ∨ 𝑥 = ∪ 𝐹)))
98	97	expimpd 453	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ((𝑥 ∈ (𝐹 ∪ {∅}) ∧ 𝑥 ∈ (Clsd‘(𝐹 ∪ {∅}))) → (𝑥 = ∅ ∨ 𝑥 = ∪ 𝐹)))
99		elin 3914	. . . . 5 ⊢ (𝑥 ∈ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ↔ (𝑥 ∈ (𝐹 ∪ {∅}) ∧ 𝑥 ∈ (Clsd‘(𝐹 ∪ {∅}))))
100		vex 3441	. . . . . 6 ⊢ 𝑥 ∈ V
101	100	elpr 4600	. . . . 5 ⊢ (𝑥 ∈ {∅, ∪ 𝐹} ↔ (𝑥 = ∅ ∨ 𝑥 = ∪ 𝐹))
102	98, 99, 101	3imtr4g 296	. . . 4 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ∈ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) → 𝑥 ∈ {∅, ∪ 𝐹}))
103	102	ssrdv 3936	. . 3 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ⊆ {∅, ∪ 𝐹})
104	21	isconn2 23330	. . 3 ⊢ ((𝐹 ∪ {∅}) ∈ Conn ↔ ((𝐹 ∪ {∅}) ∈ Top ∧ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ⊆ {∅, ∪ 𝐹}))
105	72, 103, 104	sylanbrc 583	. 2 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝐹 ∪ {∅}) ∈ Conn)
106	4, 105	syl 17	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3048  Vcvv 3437   ∖ cdif 3895   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  {csn 4575  {cpr 4577  ∪ cuni 4858  ‘cfv 6486  fBascfbas 21281  Topctop 22809  Clsdccld 22932  Conncconn 23327  Filcfil 23761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494  df-fbas 21290  df-top 22810  df-cld 22935  df-conn 23328  df-fil 23762

This theorem is referenced by:  ufildr  23847